R. E. Edwards

Fourier Series (eBook, PDF)

A Modern Introduction Volume 1

• Format: PDF

Produktbeschreibung

The principal aim in writing this book has been to provide an intro duction, barely more, to some aspects of Fourier series and related topics in which a liberal use is made of modem techniques and which guides the reader toward some of the problems of current interest in harmonic analysis generally. The use of modem concepts and techniques is, in fact, as wide spread as is deemed to be compatible with the desire that the book shall be useful to senior undergraduates and beginning graduate students, for whom it may perhaps serve as preparation for Rudin's Harmonic Analysis on Groups and the promised second volume of Hewitt and Ross's Abstract Harmonic Analysis. The emphasis on modem techniques and outlook has affected not only the type of arguments favored, but also to a considerable extent the choice of material. Above all, it has led to a minimal treatment of pointwise con vergence and summability: as is argued in Chapter 1, Fourier series are not necessarily seen in their best or most natural role through pointwise-tinted spectacles. Moreover, the famous treatises by Zygmund and by Baryon trigonometric series cover these aspects in great detail, wl:tile leaving some gaps in the presentation of the modern viewpoint; the same is true of the more elementary account given by Tolstov. Likewise, and again for reasons discussed in Chapter 1, trigonometric series in general form no part of the program attempted.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 228
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461262084
• Artikelnr.: 44181648

Inhaltsangabe

11 Spans of Translates. Closed Ideals. Closed Subalgebras. Banach Algebras.- 11.1 Closed Invariant Subspaces and Closed Ideals.- 11.2 The Structure of Closed Ideals and Related Topics.- 11.3 Closed Subalgebras.- 11.4 Banach Algebras and Their Applications.- Exercises.- 12 Distributions and Measures.- 12.1 Concerning C?.- 12.2 Definition and Examples of Distributions and Measures.- 12.3 Convergence of Distributions.- 12.4 Differentiation of Distributions.- 12.5 Fourier Coefficients and Fourier Series of Distributions.- 12.6 Convolutions of Distributions.- 12.7 More about M and Lp.- 12.8 Hilbert's Distribution and Conjugate Series.- 12.9 The Theorem of Marcel Riesz.- 12.10 Mean Convergence of Fourier Series in LP (1 < p < ?).- 12.11 Pseudomeasures and Their Applications.- 12.12 Capacities and Beurling's Problem.- 12.13 The Dual Form of Bochner's Theorem.- Exercises.- 13 Interpolation Theorems.- 13.1 Measure Spaces.- 13.2 Operators of Type (p, q).- 13.3 The Three Lines Theorem.- 13.4 The Riesz-Thorin Theorem.- 13.5 The Theorem of Hausdorff-Young.- 13.6 An Inequality of W. H. Young.- 13.7 Operators of Weak Type.- 13.8 The Marcinkiewicz Interpolation Theorem.- 13.9 Application to Conjugate Functions.- 13.10 Concerning ?*f and s*f.- 13.11 Theorems of Hardy and Littlewood, Marcinkiewicz and Zygmund.- Exercises.- 14 Changing Signs of Fourier Coefficients.- 14.1 Harmonic Analysis on the Cantor Group.- 14.2 Rademacher Series Convergent in L2(?).- 14.3 Applications to Fourier Series.- 14.4 Comments on the Hausdorff-Young Theorem and Its Dual.- 14.5 A Look at Some Dual Results and Generalizations.- Exercises.- 15 Lacunary Fourier Series.- 15.1 Introduction of Sidon Sets.- 15.2 Construction and Examples of Sidon Sets.- 15.3 Further Inequalities Involving Sidon Sets.-15.4 Counterexamples concerning the Parseval Formula and Hausdorff-Young Inequalities.- 15.5 Sets of Type (p, q) and of Type ?(p).- 15.6 Pointwise Convergence and Related Matters.- 15.7 Dual Aspects: Helson Sets.- 15. 8 Other Species of Lacunarity.- Exercises.- 16 Multipliers.- 16.1 Preliminaries.- 16.2 Operators Commuting with Translations and Convolutions; m-operators.- 16.3 Representation Theorems for m-operators.- 16.4 Multipliers of Type (LP, Lq).- 16.15 A Theorem of Kaczmarz-Stein.- 16.6 Banach Algebras Applied to Multipliers.- 16.7 Further Developments.- 16.8 Direct Sum Decompositions and Idempotent Multipliers.- 16.9 Absolute Multipliers.- 16.10 Multipliers of Weak Type (p, p).- Exercises.- Research Publications.- Corrigenda to 2nd (Revised) Edition of Volume 1.- Symbols.